I know that the only n-spheres that are Lie groups are $S^0$,$S^1$, and $S^3$. However, if I relax the condition on the points (x,y,z) of the 2-sphere to have a radius less than or equal to one, making a closed 3-ball, can the resulting set be a Lie group?
I tried to define $w=1-x^2-y^2-z^2$ so that $x^2+x^2+z^2+w^2 =1 $ and the points (x,y,z,w) are in $S^3$. However, because $w$ and $-w$ give the same point of the 3-ball, there is no map back from $S^3$ and I can't use its multiplication as the one for the 3-ball.
If that answer exists, I'm also interested in the general case for seeing n-balls as groups.